Properties

Label 68354.f
Number of curves 2
Conductor 68354
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("68354.f1")
sage: E.isogeny_class()

Elliptic curves in class 68354.f

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
68354.f1 68354i2 [1, -1, 1, -550339173, -4969145682177] 1 21282464  
68354.f2 68354i1 [1, -1, 1, 621237, -226146597] 7 3040352 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

The elliptic curves in class 68354.f have rank \(1\).

Modular form 68354.2.a.f

sage: E.q_eigenform(10)
\( q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} + q^{8} + 6q^{9} - q^{10} + q^{11} - 3q^{12} + q^{13} + q^{14} + 3q^{15} + q^{16} + 4q^{17} + 6q^{18} + 6q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.